Interval Rearrangement Ensembles

TL;DR

This paper introduces interval rearrangement ensembles (IRE), a generalization of interval exchange transformations, expanding their scope and linking them to flows on flat translation surfaces through a natural duality.

Contribution

The paper presents the concept of IREs, extends Rauzy–Veech induction to them, and explores their relation to flows on translation surfaces, offering a new framework in dynamical systems.

Findings

IRE generalizes IETs with a natural duality.

Rauzy–Veech induction applies to IREs.

IREs relate to flows on translation surfaces.

Abstract

We introduce a new concept of interval rearrangement ensembles (IRE), which is a generalization of interval exchange transformations (IET). This construction expands the space of IETs in accordance with the natural duality that we pinpoint. Induction of Rauzy\,--\,Veech kind is applicable to IREs. It is conjugate to the reverse operation by the duality mentioned above. A natural extension of an IRE is associated with two transversal flows on a flat translation surface with branching points.